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A070079
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a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).
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14
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3, 5, 15, 21, 35, 9, 45, 11, 55, 39, 65, 99, 91, 15, 105, 51, 85, 165, 19, 95, 195, 221, 105, 209, 255, 69, 115, 231, 285, 25, 75, 175, 299, 225, 275, 189, 325, 399, 391, 29, 145, 351, 425, 261, 459, 279, 341, 165, 231, 575, 465, 551, 35, 105, 609, 315, 589, 385, 675
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OFFSET
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1,1
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COMMENTS
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Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values y^2 - x^2.
Odd legs of primitive Pythagorean triangles with unique (prime) hypotenuse (A002144), sorted on the latter. Corresponding even legs are given by 4*A070151 (or A145046). - Lekraj Beedassy, Jul 22 2005
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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FORMULA
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a(n)=A079886(n)*A079887(n) - Benoit Cloitre, Jan 13 2003
a(n) is the odd positive integer with A080109(n) = A002144(n)^2 = a(n)^2 + (4*A070151(n))^2, in this unique decomposition into positive squares (up to order). See the Lekraj Beedassy, comment. - Wolfdieter Lang, Jan 13 2015
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EXAMPLE
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The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
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MATHEMATICA
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pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y^2 - x^2 /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A070079 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)
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CROSSREFS
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Cf. A002144, A002330, A002331, A080109, A070151
Sequence in context: A165260 A201874 A059528 * A142717 A057742 A201433
Adjacent sequences: A070076 A070077 A070078 * A070080 A070081 A070082
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KEYWORD
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easy,nonn
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AUTHOR
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Lekraj Beedassy, May 06 2002
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EXTENSIONS
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More terms from Benoit Cloitre, Jan 13 2003
Edited: Used a different name and moved old name to the comment section. - Wolfdieter Lang, Jan 13 2015
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STATUS
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approved
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